Where there tribes still in the 'Stone Age' in the Classical period in Europe/Asia? Did any of the great civilisations meet them and interact with them, or at least know about them?

I think it is worth quickly trying to conceptualize what "Stone Age tribe" would mean in this context, because it isn't really a term with much descriptive heft, but I think I get what you mean. To try to get at something a bit more concrete, I'll use a list of features that I'm going to be following, and if you were thinking of something else I'd be happy to adjust my overall answer:

Political independence, also can be thought of as "low levels of social/political complexity". Basically if there is a Roman or Persian law court in the village and tax collectors dropping in once a year they don't qualify, they should be relatively free in their social activity.
Low technical complexity, which is a bit difficult to define in the pre-industrial period but we can basically think of it as the number of people required in given productive activity. A Shan stilt house required as much technical ability on an individual level as the Colloseum, but the former can be feasibly built by a single family.
Individualized food production, as while I don't think we need to limit these to hunter gatherers there also shouldn't be mass plantations.
Dramatic cultural difference, which is a bit subjective but there isn't much of a way around that.

The basic answer is no for Rome, while in China there were "Stone Age tribes" until quite late and they still exist in certain more tucked away corners of India. I think a simple geographic model might explain this: China has a long and important coastline, but its empire was fundamentally inland, along the great river systems of the Yellow and Yangtze. The upshot of this is that there were large areas of interior, some of it extremely difficult to get to and thus they remained more cutoff from the currents of mainstream Chinese culture (which doesn't mean they were "isolated" or did not have complex interaction spheres of their own, they were just more difficult to bring into the Chinese ones). India is similar, with some really intimidating mountainous jungle areas. Rome, on the other hand, was geographically a coastal area, with the vast majority of the population living near the Mediterranean. Its geography lent itself to vast networks of interconnection that existed well before the Roman empire brought them under one political umbrella. It may also be the case that the climate of the area didn't lend itself to the sort of "impenetrable forests" of Yunnan or Chhattisgarh, but I couldn't say that with much confidence. This isn't to say that there were no difficult areas in the Roman empire, there were places like the Apennines in central Italy and Isuaria in southern Turkey and the eastern desert of Egypt, but these tended to be not so isolated, and were small enough that they lived in fundamental interrelation with the more mainstream, less isolated areas. The Blemmyes of eastern Egypt, for example, were considered (by Pliny at least) a strange and barbaric alien people and the name survived as a Medieval term for "marvelous humans" like the people without heads, and the Romans never even really tried to bring them to political heel. But they were far from isolated, as they participated in every capacity with the Red Sea trade, as merchant guards, raiders, and even as inhabitants in the towns (or at the very least, their agricultural styles were adopted by the port cities). There were areas within the empire that would be considered barbarian or even independent, but to paraphrase Greg Woolf, there were no "hidden valleys" of Iron Age peoples living their lives blissfully free of the dominating Roman culture.

China also had zones of lack of control within the empire, but they presented a very different practical and conceptual challenge to Chinese administrators. While the Romans wrote off the Blemmyes or Isaurians as "bandits", in China similar groups were seen as "inner barbarians" to contrast to the "outer barbarians" that were not within the empire's boundaries. These were groups like the Miao of the southwest (the modern term was much broader than the modern one) or the Li of Hainan. These were, in turn, thought of as either "raw" or "cooked" insofar as they had fallen under the civilizing influence of Chinese culture (this was often thought of as a gradient, although could be dichotomized or enumerated). This spectrum might be conceptualized as either a progress towards civilization or a degradation of noble savagery: descriptions of "cooked" barbarians being ruffians who were the worst of both worlds exist alongside more positive portrayals. The actions of the state towards them were similarly varied. The Miao of Sichuan and Yunnan have received a lot of scholarly attention because of the drama of the frontier interaction, beginning with the Song dynasty economic development of Sichuan east of the Chengdu plain leading to a great deal of conflict that could never be settled, followed by an inconsistent Ming dynasty policy of ethnic cleansing that ended when the Ming simply built a giant wall, and then the final push of the Qing dynasty that more or less fully pacified the frontier after enormously bloody struggle.

But the drama of big events obscures the more everyday interaction. The Ming and the Qing both launched bloody campaigns against he Miao, but they also recruited them quite heavily for use in their armies, treating them a bit like how the British treated "martial races" like the Gurkhas. Even more mundanely, there was never a time when trade and exchange stopped, and the Chinese who lived near the frontier were just as liable to become "raw" as the Miao were to become "cooked"--in fact the "cooking" process was seen as something requiring direct intervention of the state to prevent the frontier from becoming a lawless hell in earth of bandits and savagery.

There is also the question of origin--the traditional story is the "civilization" spreads inexorably, and barbarians are merely those who have not yet been reached by it. But there may be a flipped side: "rawness" may be a social strategy consciously adopted to preserve independence and freedom--to avoid the Chinese tax collectors, some flee to the hills. The Wa of Yunnan are a particularly interesting example of this, as they were surrounded by Chinese, Thai and Burmese empires and so developed an elaborate system of independence, including mythological systems that mocked and devauled the lowlanders. So what seem at first glance to be "Stone Age" relics are in reality anything but.

I can't really think of any comprehensive sources for this, but for the Blemmyes see Steven Sidebotham's The Red Land, for the Miao of the Song see Richard von Glahn's Country of Streams and Grottoes and for the last paragraph see James C Scott's Art of Not Being Governed.